Arithmetic nth term

Arithmetic sequences are one of the simplest patterns in mathematics, and the ability to jump directly to any term is a foundational skill for HSC Mathematics. In this problem, we practice using the explicit formula for the $n$th term to find $a_{30}$ given the first term and common difference. Mastering this technique will serve you well when tackling more complex series questions.

Problem Statement

An arithmetic sequence has first term $4$ and common difference $5$. Find $a_{30}$.

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Solutions

To find the 30th term, we apply the formula for the $n$th term of an arithmetic sequence, which links the first term, the common difference, and the term number. Since we want the 30th term, we set $n=30$; the number of steps from the initial term is $n-1=29$. Substituting $a=4$ and $d=5$ into $a_n = a + (n-1)d$ gives:

$$a_{30}=4+29\cdot 5=149.$$

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Takeaways

• The $n$th term of an arithmetic sequence with first term $a$ and common difference $d$ is $a_n = a + (n-1)d$.
• Direct substitution of the given values yields the answer without needing to list terms step by step.
• Always check that you multiply the common difference by $(n-1)$, not by $n$.

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Further Readings

HSC Last Resorts, HSC Complex Numbers, HSC Collections, HSC Integrals